Question

you are given the graph of \\(f(x) = 2^x\\).

identify the following features:

1. domain
2. range
3. \\(y\\)-intercept
4. asymptote
5. end behavior

Explanation:

Identify the domain

The function is \(f(x) = 2^x\).
The exponent \(x\) can be any real number.
Thus, the domain is all real numbers.
In interval notation, this is written as \((-\infty, \infty)\).

Identify the range

Since \(2^x > 0\) for all real \(x\), the outputs are strictly positive.
Using the Real Numbers concept, the range consists of all positive real numbers.
In interval notation, this is written as \((0, \infty)\).

Find the y-intercept

The \(y\)-intercept occurs where \(x = 0\).
Evaluating the function:

$$f(0) = 2^0 = 1$$

The \(y\)-intercept is at the point \((0, 1)\).

Determine the asymptote

As \(x \to -\infty\), \(2^x \to 0\).
The graph approaches but never touches the \(x\)-axis.
Thus, the horizontal asymptote is the line \(y = 0\).

Describe the end behavior

We analyze the behavior of \(f(x)\) as \(x\) becomes extremely large or small:
As \(x \to \infty\), \(f(x) \to \infty\).
As \(x \to -\infty\), \(f(x) \to 0\).

Answer:

Question 1

The domain is \((-\infty, \infty)\) (all real numbers).

Question 2

The range is \((0, \infty)\) (all positive real numbers).

Question 3

The \(y\)-intercept is \((0, 1)\).

Question 4

The asymptote is the horizontal line \(y = 0\).

Question 5

The end behavior is:
As \(x \to \infty\), \(f(x) \to \infty\).
As \(x \to -\infty\), \(f(x) \to 0\).